A random sample of data has a mean of 75 and a variance of...

A random sample of data has a mean of 75 and a variance of 25.
a. Use Chebyshev's theorem to determine the percent of observations between 65 and 85 .
b. If the data are mounded, use the empirical rule to find the approximate percent of observations between 65 and 85 .

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Key Concepts

Chebyshev's Theorem

Chebyshev's Theorem is a statistical rule that applies to all data distributions, regardless of shape. It states that for any k > 1, at least 1 - 1/k² of the data values will fall within k standard deviations of the mean. This theorem is particularly useful when the distribution is not known to be normal, as it provides a conservative minimum percentage of data within any chosen range.

Empirical Rule

The Empirical Rule is a guideline that applies specifically to bell-shaped, roughly normal distributions. It states that approximately 68% of the data falls within one standard deviation of the mean, about 95% falls within two standard deviations, and nearly 99.7% falls within three standard deviations. This rule is helpful for quick approximations of data spread when the distribution is symmetric and unimodal.

Variance and Standard Deviation

Variance and Standard Deviation are measurements of dispersion in a data set. Variance quantifies how much the data values differ from the mean by averaging the squared differences, while the standard deviation is the square root of the variance, providing a measure of spread in the same units as the data. These concepts are fundamental for applying both Chebyshev's Theorem and the Empirical Rule, as they define the distance from the mean in terms of standard deviations.

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